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09'2:,-f>1ETHODS FOR ROUTING A VEHICLE ON A "BIPARTITE
GRAPH AT MINIMUM COST
by
Maria Tereza Nunes Chaves de Almeida
Submitted for the degree of Ph.D.
London School of Economics and Political Science
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CUKAAI ABSTRACTIn this thesis heuristic and exact algorithms are developed to find the minimum cost of routing a vehicle on a bipartite graph subject to constraints on the number of times each node is to be visited.
The heuristic method is composed of a construction step which provides an initial feasible solution, followed by an improvement step that attempts to derive'a lower cost solution from the initial
one.
The exact method is of branch and bound type, using Lagrangean relaxation. A specialized method is used to generate .the constraints to be relaxed, to compute values for their multipliers and to update the solution, taking advantage of the particular structure of the problem.
A dynamic programming approach is also investigated, using state space relaxation and a penalty method to improve the bound. A
comparative study of the computational results obtained on a set of test problems was not favourable to this approach.
Computational experience is reported for all methods on euclidean and randomly generated cost matrix problems.
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CUKAAN ACKNOWLEOOEHENTSI am very grateful to Prof. A. Land for her supervision and incentive.
I should like to thank Dr. N. Christofides for suggesting this topic to me.
During the time I was doing this research I had the financial support of Fundayao Calouste Gulbenkian, Institute Superior de Economia (Universidade Tecnica de Lisboa) and ORS Award Scheme for which I am most grateful.
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LIST OF CONTENTS Chapter 1 - Introduction • • •. .
. . .
. . . .
.
850
Chapter 2 - Heuristic Solution • •
. .
.
.
. .
.
.
• • Chapter3 -
Lower Bounding Procedure • • • • • • • • •75
Chapter
4 -
Branch and Bound Algorithm. . .
.
. . .
• 109 Chapter 5 - D,ynamic Programming Approach • •.
. . .
• 144
Chapter6 -
Conclusions • • • •.
.
.
. . .
. .
17.2 References • • •.
. .
. . . .
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• •.
. .
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.186 Appendix •. .
.
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. . .
.
• •.
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.
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LH:l'l' OF r,IGUHES Figure 1.1 l~ 5 Fit:,'Ure 1.2 46 Figure 1.3 46 Figure 1.4 47 Fi[:>,Ure 2.1 5U Fit:,1lre 2.2 59 fie,ure 2.3 5Y Figure 2.4 60 "Figure 2.5 • 61 Ficure 2.G • 61 Fi1:,rure 2.7 63 Figure 2.8 67 Figure2.9
69
.Figure 3.1 99 Figure 3.2 100 Figure 3.3 101 Figure 3.4 • 102 Figure 3.5 10l+ Figure3.6
105 Figure lt.1 • 119n
Figure 4.2 122 Figure 4.3 122n
Figure 5.1 152 Figure 5.2 • • • • • • 159 CUKAAO-
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Figure5.3 • • • • • • • • • •
. . .
.
l''igure5.4 • • • • • •
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. . .
f1 ____
. ---
-~---~---~-1
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• CUKAAO.
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CHAPTER ONE 8 -CUKAAOu
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INTRODUCTIONIn this thesis an attempt is made to develop efficient heuristic and exact methods for the solution of a routing problem on a bipartite graph.
The designation 'routing problem' is taken here in a broad sense, that includes, generally speaking, all those problems for which the demand for service at several locations on a network is given and the optimal configuration for one or more vehicle routes is asked for.
Within this wide range of problems several categories can be established according to different criteria, namely
Cl. (a) single depot problems (b) multiple depot problems C2. (a) single vehicle problems
(b) multiple vehicle problems (b
1) one vehicle type (b2 ) multiple vehicle type C3. (a) capacity constrained problems
(b) capacity unconstrained problems
c4.
(a) deterministic demand problems(b) stochastic demand problems C5. (a) deterministic cost problems
f1
(b) stochastic cost problemsn
l
BMt-1AAA
c6.
(a) problems with the demand located on the nodes-u
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BMMAAA(b) problems with the demand located on the edges (c) mixed problems
C7.
(a) problems on directed graphs (b) problems on undirected graphs (c) problems on mixed graphsThis classification lists some important features but is by no means exhaustive. Among these categories other subcategories can easily be established. For instance, in C2 some special type of compartmentalized vehicles might or might not be considered, in C4
(a) the capacity constraints might be different from vehicle to vehicle or might be the same for all vehicles in the fleet, etc. On
the other hand, some other criteria may also be contemplated for classification purposes: the nature of the costs involved (fixed or variable), the objective pursued (minimization of total distance, minimization of total cost, maximization of utility, etc.), the nature of the operations carried out (collections only, deliveries only or both), etc.
So far, only 'pure' routing problems, i.e. problems without any temporal restrictions have been mentioned. Nevertheless the occurrence of problems subject to both spatial and temporal
constraints is quite frequent. These are designated, in this work, as mixed routing/scheduling problems. Among the temporal constrained problems it is also possible to find different types such as
Tl. problems subject to time windows
T2. problems subject to definite time service 10
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BMMAAAT3. problems·subject to maximum route time T4. problems subject to precedence conditions
Routing problems represent only one type, among others, of the problems faced by those in charge of planning departments of entities like private companies, local authorities running school bus or
household waste collection services, public transport boards, etc. Despite the strong interaction existing among routing problems and others like facility location or fleet size decision making, to mention just two, it is not rare to find situations in which the routing problems are not tackled until decisions have been made about the other issues. This is the view taken in this work.
The importance of this particular type of problems as measured by the huge amounts of money involved is well illustrated by Bodin et al.
[9]
with figures for public and private sectors both in the United Kingdom and the United States of America. These costs tend naturally to rise with the continuous increase in fuel prices.These figures and trends are certainly responsible, to a great extent, for the attention focussed on this area by so many researchers in different countries. In what follows a brief overview is taken of some of their work. Examples of problems in different categori~s
according to the above classification are given together with a brief description of the methods used for their solution.
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BMJ.1AAA1. The travelling salesman problem and related problems 1.1 The travelling salesman problem
The travelling salesman problem (TSP) is possibly the simplest routing problem to state: given a set of .n cities find a tour
beginning and ending at city 1 that visits each city exactly once and minimizes the total transportation cost. If the inter-city distance matrix is symmetric the TSP is said to be symmetric; otherwise it is said to be asymmetric.
The TSP is not a difficult problem to formulate in mathematical programming terms, either.
The symmetric version is presented in Held and Karp
[34]
asMin
s.t.
r
c.j l<i<j <n J.xij
r
xij +r
xij = 2 i=l, ••• ,n j>i j<ir
X •• (lsi
- 1 for any properHS,j"=S l.J subset S c:: { 2, ••• n
l
i<j 0 < x.j < 1 and integer .J. ld<j<n(1)
(2) (3)(4)
Subtour elimination constraints (3) can equivalently be stated in the form of
r
r
it:S JIS j>i x .. + l.Jr
r x
1J ) 2Sc{ 1, ••• ,n} S:f;.e
( 3 I ) if.S jt:S j>i 12-u
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BMMAAAThe asymmetric version is presented in Balas and Christofides
[41
asMin l: l: cij xij
iUl jtN ( 5) s.t. l: x.j = 1 i~N j(..N ~
(6)
l: xij = 1 j~N it.N (7) x .. t {O,ll ~J i,jt.N ( 8) x is a tour(9)
These are not the only nor the first formulations to appear in the literature. They were selected here because of their association with two of the most efficient algorithms developed so far for the
directed and the undirected case, respectively.
Earlier, in
1954,
Dantzig, Fulkerson and Johnson[221
formulated the TSP as in(5) - (9).
The mathematical expression given toconstraints
(9)
vas1: 1: x .. (
lsi -
1i~s jE:S ~J
SeN S;t~
(10)
In
1960
a formulation using continuous and integer variables vas presented by tUller et. al. [49].
This formulation does not seem to have been used as the basis of any particularly successful algorithm but in1979
Gavish and Graves[28)
returned to it to present two new formulations. These new formulations are shown to have a dualrelationship with Miller's and are mainly intended to be easily
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BMMAAAextended to related scheduling problems (see sections 2, 3 and 4 below).
Although it is an easy problem both to state and to formulate it is not at all an easy problem to solve~ On the contrary, as proved by Karp [39], it belongs to the difficult class of NP-complete problems.
Due to the difficulty in finding its exact solution, many
approximate methods have been proposed through the years. If the cost coefficients satisfy the triangle inequality heuristics with bounded worst-case p~rfor.mance are available [53), l54]. Some of them are construction procedures which use a particular criterion to pick up arc after arc until a tour is formed. This is the strategy followed by Clark and Wright's savings method I. 19
J
and by several insertionmethods (nearest insertion, cheapest insertion, farthest insertion, etc.). Some are improvement procedures in the sense that they begin with a feasible tour and try to improve it by interchanging some arcs. This is the strategy followed by the r-optimal methods, Lin [44j, namely by the 2-optimal and the 3-optimal methods, which interchange two and three arcs at a time, respectively. For r>3 these methods become too heav,y in terms of their computational requirements. ~bre
recently Christofides il2j developed a new construction algorithm based on shortest spanning tree, matching and eulerian tour
algorithms. He proved this heuristic to have a worst case behaviour for symmetric problems bounded by 3/2 of the optimal value. This result was refined, for n)3, by Cornuejols and Nemhauser j20i· Frieze, Galbiati and ~Bffioli [261 developed a modification of
Christofides' heuristic for which the worst case behaviour is 3xa/2,
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where a is the smallest value that satisfies the condition
i ,jtN
on the arc lengths lij•
For a brief description of sixteen heuristic methods for the TSP see Bodin et. al.
[9].
Their survey also contains the values for the worst case performances, when known, assuming the cost matrix issymmetric and the triangle inequality holds. For a comparative study of computational results using different heuristics on problems ranging in size from 25 to 100 cities see Golden et. al. [32].
A lot of effort was and still is devoted to the development of efficient exact algoritmhs for the TSP, as well. More than that the TSP has been taken as sort of a 'laboratory' where some techniques are tested. When successful on the TSP these .new techniques are then
generalized to the solution of other hard problems. A good example of that are the branch and bound methods. The designation itself appeared for the first time in the literature in Little's paper [
4·5)
on the TSP(see chapters 2 and
4).
The whole set of exact methods for the solution of the TSP can be divided into two broad categories:
(a) branch and bound methods (b) LP-based methods
As mentioned earlier the formulations presented above for the symmetric and asymmetric versions of the TSP were selected due to their association with two successful algorithms. Both are branch and
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BMMAAA1
bound type.Held and Karp
[34]
use a shortest spanning tree (~ST) relaxation approach to the TSP. They first introduce the concept of '1-tree': a graph consisting of a spanning tree on vertices {2,3, ••• ,nl plus two edges incident with vertex one. Having observed that a TSP tour is a 1-tree in which each vertex has degree two they rewrite program( 1)- ( 4) as Min s.t. I: c. j xij l<i<j <n ~ ~ x.j + ~ x.
=
2 (;-2 3 n) l. l. j .... - ' ' ••• ' j>i ~ j <i l. I: x.=
n l<i<j <n J.j I: i<::s
,j f.s
i<jX ij ... "'
lsi -
1 for any propersubset S c {2, ••• ,n l 0 < xij< 1 and integer ld<j<n
(1) (11) (12) (13) (3)
(4)
Dropping constraints (11) the problem becomes that of finding the minimum cost 1-tree, which no longer is NP-complete. Rather than being dropped, constraints (11) are carried into the objective function in a Lagrangean fashion (see chapter 3). Additionally in a second part of their paper [35], subgradient optimization is used to compute new values for the multipliers. Held and Karp's algorithm proved very efficient for symmetric problems providing a remarkable increase inu
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the size of the problems for which an optimal solution had been previously obtained. It is less efficient for the solution of
asymmetric problems both because finding a SST of a directed graph is computationally more difficult than finding a SST of a undirected graph and because the bounds obtained are poorer (see Christofides
[13]). Some variants of this method were·successively developed by Hansen and Krarup [33], Smith and Thompson
l57j
and Volgenant and Jonker l59j. Their versions differ from the initial version in the branching rules used during the tree search and in the updating formulas for the determination of the Lagrange multipliers at each node of the search tree. This algorithm is also a good example how new techniques first tested on the TSP were later on extensively used in all sorts of mathematical programming problems. In this case the Lagrangean relaxation and the subgradient optimization after being successfully used by Held and Karp on the solution of the TSP became a general tool for mathematical programming.Balas and Christofides
l
4j use an assignment problem (AP)relaxation together with a Lagrangean objective function. They first consider the AP defined by
(5) - (8)
above, then introduce some of the constraints that define the convex hull of all tours, in the formt t E
r
a;J· X;j) ao itN jd~ ... ...(14)
and relax them in a Lagrangean fashion. Rather than using subgradient optimization to update the values for the multipliers they take a
'restricted Lagrangean approach', i.e. compute these values subject to some side constraints, using up to six lower bounding procedures.
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The initial solution is kept optimal for the updated costs throughout the procedure and at the end the new reduced cost matrix is subject to an algorithm for the determination of a Hamiltonian circuit. The bounds obtained are on average closer to the optimum value for asymmetric problems than for symmetric problems and this makes the· algorithm more efficient for the former (see Christofides
[13)).
The AP relaxation with the TSP cost function has also been used in some other algorithms, like Little et. al.
[45),
Bellmore and Malone[8),
Smith, Srinivasan and Thompson[56j,
Carpaneto and TothL 10], etc. The AP relaxation usually performs better for asymmetr.ic
problems than for symmetric problems. This is explained by Balas and Toth
[5j
by the frequent occurrence of loops of length two in the latter. However, as pointed out by MiliotisL47j,
if formulation(1) - (4)
is used these loops are automatically ruled out by the fact that whenever a variable x .. is considered, variable xj. is not.~J ~
Christofides
:.13j
suggests that for the symmetric case the replacement of the AP by a matching problem leads to better results. For more details about branching rules and o~her features of some branch and bound methods for the TSP using these relaxations see Balas and Toth1 5.: e
A rather different approach is made by Christofides et. al. 15·,
using dynamic programming and state space relaxation. Departing from the observation that routing problems in general, and the TSP in particular, are essentially shortest path problems subject to some additional constraints a dynamic programming formulation is stated. To avoid the big cardinality of the state space a separable mapping
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BMMAAAfunction is used to map the state space into a lover dimensionality space - the relaxed space. The solution of the dynamic recursion on the relaxed space provides a lover bound which can be used in a branch and bound algorithm as any other bound (see chapter
5
for more details about state space relaxation and its application to the solution of the TSP and the MCBCP).· Attempts have also been made to solve the TSP using LP- based methods. Miliotis 147:: developed a method for the symmetric case based
on formulation (1) -
(4) •
The algorithm uses tvo types ofrelaxations: the relaxation of integrality constraints as usual and the relaxation of part of the other constraints that define the problem. The first goal is to achieve an integer solution. For the variables that violate integrality constraints a branching, using Little's strategy, is performed. When an integer solution is achieved it is checked for feasibility. If not feasible some of the initially omitted and currently violated constraints are generated and
introduced and the problem reoptimized. The nUmber of constraints generated until a feasible, and therefore optimal, solution is
obtained is only a small part of the whole set of constraints defining the TSP. In Miliotis ,48, tvo new versions of this algorithm are
presented. The basic approach is kept but to achieve an integral solution cutting planes are used rather than branching. The tvo versions, identified as straight algorithm and reverse algorithm, differ in the order in which the cutting planes and the omitted
constraints are added. In all versions the tests for feasibility and the generation and introduction of nev constraints are done using Land and Powell 4o, codes. The computational results presented indicate
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that for large size problems the reverse algorithm is more efficient but for average size problems they seem to be equivalent. Once more, after testing this method on the TSP the author suggests that it is likely to be efficient in the solution of other integer programming problems.
Another LP based algorithm is described in Crowder and Padberg 21 .• The formulation differs from (1) - (4) in that constraints (3) are replaced by the so-called comb constraints
k E i=O x(W.)
<
IW I ~ 0 k + L i=l(lw
.1 -
1 ) - <1/2 x k> ~ ( 3' I )where<.> stands for the next highest integer and W. (i=O, ••• ,k) ar.e
~
proper subsets of the node set for which conditions
lw fl w.l ) 1 i=l, ••• ,k 0 l. (15) lw. - w I ) 1 i=l, ••• ,k l. 0
(16)
IW. ll W.l = 0 . l<i<j <k l. J (1 7) k odd (18)hold. For each of the subsets, W., x(W.) is defined as
l. l.
x(W.) J.
=
.r
x e x EE(W.) . e J.(19)
where x are the components of a vector associated with the edge set E
e
and E(W.) is the subset of edges with both ends in W .• The method uses
l. l.
an heuristic solution to start a primal linear programming iterative procedure that first include~ only constraints (2) and successively
BMMAAA
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B~1MAAAl
generates new constraints, in order to get a tighter relaxation. If before reaching optimality a situation is faced where no new
constraints can be generated the linear programming problem is solved to optimality and a true lower bound is established. If possible, without destrqying primal feasibility some subtour elimination constraints are appended to the linear problem. Alternatively a branching strategy is also presented. The method is reported to solve problems up to 318 cities. The authors explain its effectiveness by the fact that the cutting planes introduced define facets of the polytope underlying the TSP and are therefore quite strong.
1.2 The travelling salesman problem with time windows
The TSP with time windows is an ordinary TSP except that each city can be visited only during one or more intervals• A processing time for each city might or might not be considered and the salesman might or might not be allowed to wait, if he arrives at a city outside
its time windows.
Baker
[21
considers that associated with each city i there is only a time window,[1.,
u.], and the salesman is al~owed to wait, if~ ~
necessary, for the opening of a time window. The distance matrix d is assumed to be symmetric and satisfying the triangle inequality.
Representing by t
1 the moment that the salesman arrives back to city n+
1, the problem is formulated as
Min (20)
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BMl'-1AAAl
s.t. t. -t ) dli i=2, ••• ,n (21) ~ 1 lti -tjl )dij i=3, ••• ,n; 2<j<i (22) t n+ 1 -t. ~ ) dil i=2, ••• ,n (23) t.> 0 i=l, ••• ,n+l (24)
~
1. ( t. ( u. i=l, ••• ,n+l (25)
~ ~ ~
A branch and bound method is proposed to solve it. The method is based on the fact that the dual of the relaxed problem obtained by relaxing constraints (22) and (25) is a longest path problem in a directed graph with n+l nodes.
Christofides et. al. [15] extend their dynamic programming approach to the TSP to a slightly different version of the TSP with time windows in which a processing time, t5., and
~ ri ordered and
<: k. < r. are disjoint time windows in the form
[e~i,
~
u~i],
~ 1 ~ ~associated with each city. The salesman is also allowed to wait, if necessary, for the next time window to open. Therefore if city i is visited immediatly after city j one of the next three conditions must hold t.= t.+ ~ t.= ~ t.=oo ~ J k e. ~ 0. + c .. if J J~ if if k ( t.+ t5.+ cji < k e. u. ~ J J ~
..
k k-1 u. < t.+ o.+ c .. ( e. ~ J J J~ r. t + o.+ c .. > u.~ j J J~ ~ - 22 -~ (26) for some (27) k=l, •• ,r. ~ (28)u
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BMMAAAThe dynamic programming recursion is adapted to these new
conditions and state space relaxation is again used to provide bounds to be embedded in a branch and bound algorithm.
1.3 The multiple travelling salesman problem
The multiple travelling salesman problem (M-TSP) is a generalization of the TSP. Rather than one, a set of M routes,
beginning and ending at the depot, and visiting each customer exactlY once, are required, minimizing the total cost. A M-TSP can easily be converted into a single TSP by creating M copies of the depot, each connected to the customer nodes exactlY as the original depot, and such that no two copies are connected.
Bodin et. al.
[9]
formulated M-TSP as Min s.t. ~ ~ i j~
X • • = bJ. = ( Ml i~J
1
~
x=
a=J
M j i j ill
X=
X •• ) t 8 l.J if j=l if j=2, ••• ,n if i=l if i=2, ••• ,n x=
0 or 1 - i ,j =1~ ••• ,n i j (29) (30) (31) (32) (33)for any choice of the set S in {32) that breaks subtours which do not include the origin.
A branch and bound algorithm for the M-TSP is presented in - 23
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BMMAAAChristofides et. al.
[16]
based on a graph referred to in his work as the k-degree centre tree (k-DCT). The method is also extended to the general routing problem (see section7),
after noting that the latter is the former subject to some additional constraints. The k-degree centre tree is simply a tree graph in which the node representing the depot, say node x , has degree k. This is the kind of graph obtained0
if M arcs are deleted, one in each route, from a feasible solution for the M-TSP. Let S represent the subset of y deleted edges incident
0
with x and S1 represent the subset of M-y deleted edges not incident 0 with x • The resulting tree is a k-DCT with k=2xM-y. The arcs in the
0
M-TSP tour are partitioned into three subsets: S , S1 and the k-DCT.
0 Let
=l:
if arc l(S F.o 0 1 otherwise (34) F.l=t
if arc 1~s
1 1 otherwise .(35)t
if arc l~k-DCT F.l = otherwise (36)The M-TSP can therefore be formulated as
Min (37) (38) E F.
=
2M-y lEA l (39) 0 24-u
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BMMAA.A m E ~ 1=
N 1=1 0 E ~=
y lE:A l 0 E l~A-A 0 1 ~=
M-y 1 E(~
+~0 +~1
)= 2 lEA. 1 1 1 i=l, ••• ,n J. for all 1 for all 1 for ail 1 (40) (41) (42) (43) (44a) (44b) (44c)where.~ represents the set of all arcs incident with k (k=O, ••• ,n). If constraints (43) are relaxed in a Lagrangean fashion, for each value of y the problem is decomposable into three smaller problems
P
1- defined by constraints
(38), (39),
(40) and (44a) P2- defined by constraints (41) and (44b)P
3- defined by constraints (42) and (44c)
If their Lagrangean objective functions are denoted by v(A,y), vO(A,y) and vl(A,y), respectively, the best lower bound is given by
{Max [v(A,y) + vo(A,y) + vl(A,y)]
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where M
1 is a problem specific value •.
These lower bounds can be embedded in a branch and bound
]
algorithm as usual.n
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1.4 The clustered travelling salesman problemn
The clustered travelling. salesman problem (CTSP) is the standardn
TSP with the additional condition that the cities are pmrtitioned into groups and each group of cities (cluster) must be visitedn
contiguously.n
This is the situation when operations with differemt tools must be performed at several points on a plane surface. As tool changingu
takes a long time all operations with the same tool (cbs.ter) must be performed before another tool is put into the machine.n
Jongens and Volgenant [ 38) studied the symmetric cue. Their ·IT
approach is based on the algorithm developed by Volgenrumt and Jonkern
[59]
for the symmetric TSP. They consider the set of e~s divided into two subsets: the subset of edges joining nodes belmnging to then
same cluster, called local edges, and the subset of ed~ joining nodes belonging to different clusters, called nonlocal edges. Any0
feasible solution must contain as many nonlocal edges as the number ofu
degree constraints as in clusters in the problem. This condition is, together with the node[59],
relaxed in a Lagrapgean ffashion ton
obtain a lower bound on the optimal value of CTSP. It is suggested that a variant consisting of introducing as many multipmiers as then
number of clusters and relaxing the constraints on each cluster'sn
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degree might speed up the algorithm.
Taking advantage of the particular structure of CTSP three new rules were developed to include and to exclude edges from the
solution. They are used during the branch and bound step of the algorithm, together with rules previously developed for the TSP
[59]
to abbreviate the search.-[1
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2. The deliver,y problem
Gavish and Graves
[28]
present a routing problem subject to capacity constraints which they designate as deliver,y problem, as follows: giyen M vehicles, each ?f them with a maximum capacity Q, and a nonnegative load q1 associated with each node of the graph, find M
t9urs of minimum cost that leave depot n
0, collect the respective load
and return to the depot. For a graph with n customer nodes the problem is formulated using (n+l)2 integer variables and (n+l)2 continuous variables as Min s.t. n n I: I: c.j i=O j=O ~ n I: X •• = 1 i=O ~J n I: x ..
=
1 j=O ~J n I: x0J.=
M j=O n n xij j=l, ••• ,n i=l, ••• ,n I: Y .. - I: Y ..=
d. j=O ~J j=O J~ ~ y .. ( Q X •• ~J ~J ' x=
0 or 1 y .. ) 0 ij ~J(46)
(47)
(48) (49) (50) i=l, ••• ,n(51)
i,j=O,l, ••• ,n (52) i ,j =0, 1, ••• ,n (53)If the number of vehicles is not given beforehand but a fixed cost p is associated with each vehicle used, the objective function BMMAA.C 28
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shall be replaced b,y
n n n
Min I: I: cij xij+ I: (cOj+ p) xOj
i=l j=O j=O
(461) and constraints
(49) -
(50) by n n I: x0j- I: xj 0= 0 j=l j=l(54)
No method is directly proposed for the solution of this deli very problem. The formulation is presented as an extension of a formulation for the TSP (see section 1.1). For the TSP itself computationalexperience is reported for
4
problems ranging in size from5 up to
42
nodes, using two linear programming relaxations. These methods are expected by the authors to perform well also for this delivery problem.-u
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3. The school bus problem
School bus services are the kind of routing problem in which safety considerations must be included. The most popular way of taking them into account is to establish upper bounds, T., on the time,
J.
t. , each student is to spend travelling.
J.O
TYpicallY the bus leaves the school (say node n ), collects
0
students w~iting for it at n pick-up nodes and returns to the school. The number qi of students waiting for the bus at each pick up node ni is known beforehand as well as the bus capacity Q.
Gavish and Graves
[28]
formulate the problem assuming that onlY one bus is allowed to stop at each pick-up node, that the loading time at each stop is negligible and that the fixed cost of using an extra bus is p Min s.t. BMMAAC n n n E E c .. x 1j+ E ( c0j+ p) x0ji=l j=O l.J j=O
n E x ..
=
1 i=O l.J n E X •• = 1 j=O l.J n n }; yij- E j=O j=l y,. < l.J Q xij j=l, ••• ,n i=l, ••• ,n yji= d. J. i=l, ••• ,n i,j=O, ••• ,n 30-(55)
(56)
(57) (58) (59)J
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i=l, ••• ,n (60) i,j=O, ••• ,n (61) (62)This formulation is also presented as an extension of the
authors' formulation for the TSP and the remarks made for the delivery problem apply here as well.
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4.
The dial a bus problemThe dial a bus services attempt to combine the advantages of taxi services with the lower costs of the bus system. A trip is
requested by telephone specifying the collection point, the number of passengers and their destinations and the earliest and latest
allowable times. The system has to be able to make a decision: either to send a bus directlY from the depot or to redirect a bus already on route. This decision shall take into account bus capacity, limits on waiting time for each passenger and, quite frequentlY, special requests
like extra capacity for luggage, help for handicapped or old people, etc.
Gavish and Srikanth [29] present systems at various levels of complexity. First, a system with onlY one unlimited capacity bus and one passenger at each pick-up point is formulated. It is formulated as a quadratic programming problem, adapting a quadratic assigmnent
problem formulation for the TSP due to Lawler [42] and as an integer programming problem, adapting a formulation to the TSP due to Gavish and Graves [28]. The model is then adapted to accommodate more than one passenger per pick-up point, bus capacity and passenger travel time constraints. New formulati~ns are given for systems with multiple busses and one passenger per pick-up point, and for systems with
different bus types and capacity and time constraints. As area for further research the authors indicate the adaptation of existing solution methods for transportation/scheduling problems.
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5.
The chinese postman problemThe chinese postman problem {CPP) was first presented in a
chinese periodical (hence the name) and is that of finding the minimum cost route that traverses each edge of the graph at least once. For totally directed and totally undirected graphs this is an easy problem to solve as compared to the TSP and related problems
[52).
Possibly because of that, there is a generalized idea that arc constrained problems are easier to solve than riode constrained problems and some attempts have been made to transform some of the latter into some of the former. But, as pointed out by Lenstra and Rinnooy Kan[43]
the conversion of required nodes into required arcs does not essentially decrease the complexity of the problems. They also proved that the Rural Postman Problem, an arc constrained problem (see section6),
is as difficult as the TSP to solve. The CPP is the type of problem faced not only by postmen delivering post but also by farme~s seeding their fields, by track repair crews repairing networks, etc. The unde~lyinggraph ~ay be undirected, directed or mixed.
Minieka
[50), [51]
studies the three situations.For the undirected graphs two different cases are considered separately
(i) the underlying graph is even (ii) the underlying graph is not even
If the underlying graph is even, i.e. all its nodes have even degree, the postman does not have to repeat any edge and the CPP is simply that of finding an eulerian tour. The determination is made BMMAAC 33
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using a very simple algorithm for building cycles and splicing them together.
If some nodes do not have even degree then at least one edge incident with each odd degree node has to be repeated. This situation is solved using a shortest path algorithm followed by a weighted matching ~lgorithm.
Unlike the undirected CPP the directed CPP may have no solution: if there is a set of nodes S in the graph such that no arcs go from a node in S to a node not in S there is no solution at all. The postman gets trapped inside S and cannot complete the route. When a solution exists two cases are again considered
(i) the underlying graph is symmetric (ii) the underlying graph is not symmetric
If the underlying graph is symmetric, i.e. if all nodes have indegree equal to outdegree, there is no need for the postman to repeat any arc and the solution may be obtained using an algorithm similar to the one used for the even undirected case.
If the underlying graph is not symmetric then the postman has to repeat some arcs: for nodes having more arcs entering them than
leaving them the postman has to repeat some leaving arcs and for nodes having more arcs leaving them than entering them the postman has to repeat some entering arcs. The problem is solved using a minimum cost flow problem algorithm followed by an eulerian tour algorithm.
As in the directed case if the underlying graph is mixed it may BMMAAC 34
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happen that a set of nodes exists such that no way out of it can be found. In this case no solution exists at all. If a solution exists then three situations are considered
(i) the graph is even and symmetric (ii) the graph is even but not symmetric (iii) the graph is neither even nor symmetric
If the graph is even and symmetric the solution method is simply a combination of the methods used for the even undirected and directed cases.
If the graph is even but not symmetric it is not easy to assess in advance whether or not repetitions are inevitable. The mixed
postman algorithm transforms the graph into an even directed graph by arbitrarily selecting a direction for each undirected edge. This new problem is solvable by the methods already mentioned. As the choice is arbitrary some corrections are subsequently made on some arc
directions.
If the mixed graph has some odd degree nodes the problem becomes NP-complete [52]. Minieka [50] outlines a two stage approach for this case but solving each stage to optimality does not guarantee an optimal solution for the overall problem. In [51] he rephrases the problem as an integer minimum cost flow with gains problem that can be solved by integer linear programming techniques.
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6. The rural postman problem
The rural postman problem (RPP) is, like the CPP, a routing problem with the· demand located on the edges, the difference being that on~ a subset of edges, the so-called required subset, must be traversed, at least once, at minimum cost. The delivery of post or milk, the inspection of gas pipelines and electric power distribution networks are some examples of real life situations that can be
modelled as a RPP. The difficulty in solving the problem depends on the number of components formed by the required subset. If it does not form a single connected component the problem is NP-complete and can be shown to be a generalization of the TSP (see Christofides et. al.
[17], [18]). The under~ing graph may be directed or undirected. The undirected version is studied in Christofides et. al. [17]. The heuristic method proposed is composed of two stages: a
construction stage followed by an improvement stage. The first is similar to Christofides heuristic for the TSP (see section 1.1) in that it also consists of the solution of a shortest spanning tree followed by the solution of a matching problem. The latter is an attempt to improve the solution alreaqy obtained, by replacing pairs of edges by a single edge, reducing the total cost. The exact method begins with a sequence of transfonnations on the under~irig graph in order to simplify its structure and consequent~ the formulation of the problem. Lagrangean relaxation is used on some of the constraints and the resulting problem is solved by partitioning it into two simpler problems. One is solved by inspection of its coefficients and the
other using a shortest spanning tree method. Some constraints
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identified as violated are later on added to the problem and also relaxed in a Lagrangean fashion. The bounds obtained are used in a branch and bound algorithm. Computational experience is reported for a set of
24
randomly generated problems ranging in size up to84
nodes, 18o edges, 74 required edges and 8 connected components.The directed version is studied in Christofides et. al.
(18).
Anhe?ristic method is proposed which consists of the solution of a
.
.
shortest spanning tree on the condensed graph followed by the solution of a minimum cost flow problem. The same improvement procedures
ment.ioned for the undirected version are tried. In the exact method some conditions are previously established (concerning the inclusion of certain arcs} that allow simplifications on the initial graph. Once the problem is formulated Lagrangean relaxation is used on some
constraints. The resulting Lagrangean problem is decomposable into.two smaller problems. One is solved by inspection of its coefficients and the other by solving a shortest spanning arborescence problem. The
solution of the former is simplified by the conditions under which the Lagrange multipliers are computed. The bounds obtained are, as usual, embedded in a branch and bound algorithm. Computational experience is reported for 23 problems randomly generated and ranging in size up to
80 nodes, 179 arcs, 71 required arcs and 8 components.
CUSAAC
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CUSAAB7.
Vehicle routing problemThe vehicle routing problem (VRP) is an extension of the M-TSP in which a set of customers must be served ~y a fleet of .vehicles, all located at a central depot, minimizing a distribution function.
Christofides et. al.
[16]
assume that for each customer xi, i=l, ••• ,N,a
demand qi and a service cost ui are given, that each vehicle vk, k=l, ••• ,M, has a capacity Q and that an upper bound Texists on the total cost of each route.
If cij represent the travelling costs fom i to j and the
distribution function to be minimized is the total cost the problem can be formulated as Min z= s.t. N· N M E
z
(cij E~ijk)
i=O j=O k=l N M E E C" = 1 j=l, ••• ,N i=O k=l "'ijk N N k=l, ••• ,M; p=O, ••• ,N N Nr (
q.r
~. "k) < Q i=l 1 j=O 1J N E i=O N E j=l Nr
c .. j=O 1J C"=
1 "'ojk y.- y.+ N 1 J N ~ijk+ E (u. i=l 1 M E ~iJ"k < N-1 k=l - 38-Nr
~ijk)
< T j=O k=l, •• ,M i*j=1, ••• ,n (63) (64) (65) (66) (67) (68)(69)
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CUSAAB~ijk f {0,1} for .all i
,J
,k (70) yj arbitrary for all J (71) where.{i,
~ijk
=
o,
if vehicle k v~sits j immediately after i
(72) otherwise
A slightlY different formulation is presented in Christofides et. al. [14] using a more general objective function.
This is a basic formulation that does not comtemplate other conditions very frequent in real life problems, such as:
Cl- different types of products to be delivered by a vehicle C2- unloading time spent at each customer
C3- time windows during which the deliveries are to be carried out
C4- working time constraints
C5- existence of compartmentalized vehicles in the fleet C6- loading time at the depot
etc. These conditions are not easy to accommodate by formulation (63) - (71) which represents, by itself, a large and difficult integer linear programming problem.
Also in Christofides et. al. [14] a set partitioning approach is presented that can easilY accommodate extra constraints. It consists of generating the totality of the routes a single vehicle can operate. For each route the optimal tour is computed, which involves the
solution of TSPs (and can therefore, depending on the number of
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customers in each route, be quite time consuming). A matrix is then produced whose rows correspond to the feasible routes. The columns are grouped by blocks, each one corresponding to routes operated by a particular vehicle. The problem becomes that of choosing at most one column from each block so that each row is covered exactly once and the total cost is minimized. This problem can be transformed into a set partitioning problem. As the total number of feasible routes can be extremely large the time consumed on their generation and the resulting set partitioning problem may become enormous, unless very powerful dominance and exclusion tests are used early in the process to decrease significantly the number of routes considered. Another approach is presented in the same work that avoids the a priori generation of all feasible routes. This is a tree search algorithm, with a depth first strategy, in which the feasible routes are
generated as and when required. Each node on the tree corresponds to a route that can be feasibly operated by at least one vehicle. In · Christofides et. al.
[15]
the dyn~ic programming approach to the TSP is also extended to the VRP. Denoting by f(m,S) the least cost of supplying a set of t4 customers using only m vehicles and by v(S) the solution of the TSP defined by the set S of customers and the depot the following dynamic programming recursion is establishedf (m, S)
=
Min { f (m - 1, S - L) + v (L) } Lc:S s.t. E q.- (m -1)
Q < "-S ~ x ... ~ E q. < Q L L~ x.~ ~(73)
(74)[1
for m=2, ••• ,M where S must satisfyJ
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CUSAAB-I
(75) initialized by f (1, S) = v (S) (76)The constraints introduced prevent the computation for sets that cannot lead to feasible solutions. The formulation and two other
alternatives slightly different are used as the basis for an algorithm using state space relaxation to be embedded in a branch and bound.
The k-degree center tree approach [16] used to solve the M-TSP (see section 1.3) can also be used for the solution of the VRP. The additional-constraints present in the VRP can be used to derive a better value for the parameter M1• 'lhis is taken to be the largest value for which
N {M - f-1 1) Q ;> E qi i=M1+1 (77) N (M - M 1) T ;> E t. i=M1+1 ~ (78) hold.
Another suggestion for the computation of lower bounds on the value of the VRP, based on q-routes, is also made in Christofides et. al. [16]. Let W be the set of all possible loads that could exist in any feasible route
w
= { qI
E ~~i = q ( Q i for some~, ~-~{0,1}}
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CUSAABLet the elements of W be ordered in ascending order and let w=IWI. Let q(l) denote the 1th element of Wand rr(q) that 1* so that
q(l*)=q. The total load on a path ~=(x
0
,xi1
, ••• ,xik) not necessarilY simple is defined asr q. (8o)
X.E~-{x} ~ ~ 0
If f
1(x.) represents the least cost path from x to x. with load ~ 0 ~
q(l) then the route resulting from it adding arc (xi, x
0) , the
q-route, has a cost f'
1(x. )=f~ 1(x,)+c, • This value is a lower bound ~ ~0
on the cost of a VRP route. ~namic programming and Lagrangean penalties are then used to compute the values of the lower bounds.
The computational experience presented is that the bounds obtained from the q-routes are superior to those obtained from the k-DCT and that this method is capable of solving problems up to 25 nodes.
Being such a difficult problem it is ,not a surprise that several heuristic methods were also proposed. Most of them are of the
construction type but once a feasible solution is obtained it is also possible, as for the TSP and other routing problems, to apply some local optimization techniques in an attempt to improve the solution. Among the construction type methods two broad categories can be found
(i) simultaneous type (ii) sequence type
The former construct more than one partial route at a time and progressively extend them until complete routes are formed while the
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CUSAABlatter construct one route at a time until a complete sequence of routes is obtained.
For a brief description of som~ alreaqy published heuristics, as well as some pew ones (including one based on the tree search exact
algorithm mentioned above) see Christofides et. al. [14].
More recently Stewart and Golden
[58]
presented an algorithm whose computational results on a set of 8 problems picked up from the literature compared favourably to most of the previously developed heuristics. It is a composite type heuristic formed by a construction step followed by an improvement step. The construction step usesLa.grangean relaxation to transform the VRP into a M-TSP whose proposed solution satisfies the VRP's capacity constraints too. The improvement step is of 3-optimal type but allowing intermediate infeasible
solutions to be considered.
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8.
A new routing problem on a bipartite graph In this thesis a particular one-vehicle, multiple depotrouting problem is studied. This type of problem has some real world applications both in the distribution of chemicals and in the
collection of milk from fanns.
The underlying graph is bipartite as after visiting a customer the vehicle must go to a depot (either to unload the milk or to be cleaned and reloaded with chemicals) before another customer is visited. The vehicle may return to the departure depot or may go to another depot (as long as the depot capacities are.met). Only the undirected version is studied in this work, so any edge linking a depot to a customer is considered traversible in both directions. Let X={xl'x2 , ••• ,xN} be the set of depot nodes and Y={yl'y2 , ••• ,y~.
1
} be the set of customer nodes. It is also assumed thatr
d(x.).=r
d(yj) x.EX ~ yff:Y~ j
(81)
where d(.) represents the degree of node (.) in a feasible solution. Each customer must be visited exactly once and hence d(y. )=2 for all
J j. Each depot is due to supply k.=d(x.)/2 customers.
:1. ~
The problem is that of finding the minimum cost circuit that visits each customer exactly once and satisfies the depot capacity constraints. From now on this problem will be referred to in this work as the minimum_ cost bipartite circuit problem (MCBCP). A typical
feasible solution looks like figure 1 where an arbitrary 3 depot and 12 customer problem is considered.
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CUSAAB Figure 1As shown, the number of customers supplied by different depots may be different
- depot x1 supplies 5 customers - depot x2 supplies 3 customers - depot x
3
supplies4
customers.For small examples this is the most frequent pattern exhibited by optimal solutions: a simple central bipartite circuit plus some pairs of edges linking customer nodes not in the central circuit to depot nodes in the central circuit. However for larger problems it is very frequent to find solutions like the one illustrated in figure 2 where the simple central bipartite circuit is replaced by a compound :
bipartite circuit.
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CUSAAB..,.,
.
~ ~~~:~~·y1
• Yz. Figure 2Under particular conditions feasible and optimal solutions exist which do not contain circuits of length 2.
If each depot is to supply exactly one customer all depots have degree 2 in any feasible solution • In this case no connected solution containing circuits of length 2 can exist.
If the cost matrix C satisfies the condition c =k for all i and
ij
j then depending on the number of customer nodes and on the constraints on the number of customers each depot is to supply,
optimal solutions may exist which do not contain circuits of length 2. Figure 3 illustrates for a 2 depot and
4
customer problem an optimal solution without such circuits.Figure 3
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For this problem alternative optimal solutions exist as the one in figure4
Figure 4
As the conditions under which no optimal solution with circuits of length 2 exists are ver,y unlikely to occur the examples given from now on will all contain them.
In the remaining illustrations, to simplify the drawing, the number of times each edge is traversed in the solution is stated close to the edge, i.e. if an edge is traversed twice (once in each
direction) number 2 is drawn next to it rather than the edge duplicated.
In the following chapters the underlying graph is denoted by G=(V,E) where V represents the set of nodes and E represents the set of edges. The node set satisfies conditions
V =XU Y XflY=-8"
where X represents the depot node set and Y represents the customer CUSAAB 47
